Three-term recursion relations for hydrogen wave functions: Exact calculations and semiclassical approximations

Abstract

Three-term recursion relations with respect to the angular momentum l are given for the normalized hydrogen wave functions associated with r, pr, p (r is the radial polar coordinate in configuration representation, pr is the momentum conjugated to r, p is the radial polar coordinate in momentum representation). These three-term recursion relations [Eqs. (4), (6), (16)] are found numerically stable in the order of decreasing l values, even for large quantum numbers. The three-term recursion relations in r and p are used to derive semiclassical approximations for the radial wave functions Pn,l( p) and Rn,l(r). These semiclassical approximations [Eqs. (67) and (84)] are valid even at the classical turning points and are still markedly good at small quantum numbers.